Calculating the optimal exercise boundary of American put options with an approximation formula Song-Ping Zhu School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia Telephone number: 61-2-42213807, Fax number: 61-2-42214845 e-mail: spz@uow.edu.au Abstract In this paper, an algorithm to improve the computational accuracy of the analyt- ical approximation to the value of American put options and their optimal exercise boundary proposed by Zhu (2004) is presented. In the current approach, Zhu’s simple approximation formula is used as an initial guess for the optimal exercise boundary of American put options. The determination of an improved optimal ex- ercise boundary is then achieved by setting a null value of the Theta of option value on the optimal exercise boundary. Test example results show that the improvement is indeed significant. 1 Introduction It is well known now that the valuation of American options mathematically poses a free boundary problem under the Black-Scholes (1973) framework and the correctly pricing the option crucially depends on an accurate calculation of the so-called op- timal exercise boundary, which varies with time and is part of the solution. The necessary determination of the optimal exercise boundary has made the pricing of American options much harder than pricing their European counterparts; the search for accurate and efficient methods to calculate the optimal exercise boundary in the valuation of American options has been pursued by many researchers in the past two decades. In the literature, there have been two types of approximate approaches, numerical methods and analytical approximations, for the valuation of American options. Each type has its own advantages and limitations. Of all numerical methods, there are two subcategories, those with which the Black-Scholes equation is directly solved with both time and stock price being dis- cretized and those based on the risk-neutral valuation at each time step. The former subcategory typically includes the finite difference method (Brennan and Schwartz (1977), Wu and Kwok (1997)), the finite element method (Allegretto et al. (2001)) 1 380 IWIF 1, 2004, Melbourne, www.iwif.org